MathDB
Problems
Contests
National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
1994 Irish Math Olympiad
1994 Irish Math Olympiad
Part of
Ireland National Math Olympiad
Subcontests
(5)
5
2
Hide problems
inequality with functions
Let
f
(
n
)
f(n)
f
(
n
)
be defined for
n
∈
N
n \in \mathbb{N}
n
∈
N
by f(1)\equal{}2 and f(n\plus{}1)\equal{}f(n)^2\minus{}f(n)\plus{}1 for
n
≥
1
n \ge 1
n
≥
1
. Prove that for all
n
>
1
:
n >1:
n
>
1
:
1\minus{}\frac{1}{2^{2^{n\minus{}1}}}<\frac{1}{f(1)}\plus{}\frac{1}{f(2)}\plus{}...\plus{}\frac{1}{f(n)}<1\minus{}\frac{1}{2^{2^n}}
partitioned square
If a square is partitioned into
n
n
n
convex polygons, determine the maximum possible number of edges in the obtained figure. (You may wish to use the following theorem of Euler: If a polygon is partitioned into
n
n
n
polygons with
v
v
v
vertices and
e
e
e
edges in the resulting figure, then v\minus{}e\plus{}n\equal{}1.)
4
2
Hide problems
matrices
Consider all
m
×
n
m \times n
m
×
n
matrices whose all entries are
0
0
0
or
1
1
1
. Find the number of such matrices for which the number of
1
1
1
-s in each row and in each column is even.
equations
Suppose that
ω
,
a
,
b
,
c
\omega, a,b,c
ω
,
a
,
b
,
c
are distinct real numbers for which there exist real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
that satisfy the following equations: x\plus{}y\plus{}z\equal{}1, a^2 x\plus{}b^2 y \plus{}c^2 z\equal{}\omega ^2, a^3 x\plus{}b^3 y \plus{}c^3 z\equal{}\omega ^3, a^4 x\plus{}b^4 y \plus{}c^4 z\equal{}\omega ^4. Express
ω
\omega
ω
in terms of
a
,
b
,
c
a,b,c
a
,
b
,
c
.
3
2
Hide problems
polynomials
Find all real polynomials
f
(
x
)
f(x)
f
(
x
)
satisfying f(x^2)\equal{}f(x)f(x\minus{}1) for all
x
x
x
.
inequality
Prove that for every integer
n
>
1
n>1
n
>
1
, n((n\plus{}1)^{\frac{2}{n}}\minus{}1)<\displaystyle\sum_{i\equal{}1}^{n}\frac{2i\plus{}1}{i^2}
2
2
Hide problems
equilateral triangle
Let
A
,
B
,
C
A,B,C
A
,
B
,
C
be collinear points on the plane with
B
B
B
between
A
A
A
and
C
C
C
. Equilateral triangles
A
B
D
,
B
C
E
,
C
A
F
ABD,BCE,CAF
A
B
D
,
BCE
,
C
A
F
are constructed with
D
,
E
D,E
D
,
E
on one side of the line
A
C
AC
A
C
and
F
F
F
on the other side. Prove that the centroids of the triangles are the vertices of an equilateral triangle, and that the centroid of this triangle lies on the line
A
C
AC
A
C
.
find all possible values
Let
p
,
q
,
r
p,q,r
p
,
q
,
r
be distinct real numbers that satisfy: q\equal{}p(4\minus{}p), \, r\equal{}q(4\minus{}q), \, p\equal{}r(4\minus{}r). Find all possible values of p\plus{}q\plus{}r.
1
2
Hide problems
identity
Let
x
,
y
x,y
x
,
y
be positive integers with
y
>
3
y>3
y
>
3
and x^2\plus{}y^4\equal{}2((x\minus{}6)^2\plus{}(y\plus{}1)^2). Prove that: x^2\plus{}y^4\equal{}1994.
sequence
A sequence
(
x
n
)
(x_n)
(
x
n
)
is given by x_1\equal{}2 and nx_n\equal{}2(2n\minus{}1)x_{n\minus{}1} for
n
>
1
n>1
n
>
1
. Prove that
x
n
x_n
x
n
is an integer for every
n
∈
N
n \in \mathbb{N}
n
∈
N
.